JGSP 8 (2006) 123–126
BOOK REVIEW
Differential Geometry and Lie Groups for Physicists, by Marian Fecko, Cambridge University Press, Cambridge, 2006, xv + 697 pp. ISBN 978-0-521-84507-6
A modern mathematical physicist could hardly imagine doing serious studies in
the frame of contemporary theoretical and mathematical physics without making
substantial use of concepts like vector and tensor fields, differential forms, Stokes
theorem on manifolds, Lie derivative and symmetries, Lie (super)groups and Lie
(super)algebras and their representations, bundles with connection, curvature and
Chern classes, Hamiltonian and Poisson structures, Frobenius integrability. Differential topology has already also made serious steps inside theoretical physics
through topological charges, helicity properties of magnetic fields in studying
plasma properties, topological soliton theory and other homology/cohomology
characteristics of physical systems wherever they are available and important.
Any physical system has spatial structure and shows definite stability properties,
so, it can support its existence and compensate in definite degree the external disturbances through appropriate shape changes and kinematical behavior without
losing identity. Shortly speaking, its time existence is a dynamical process being strongly connected with various and continuous internal and external energymomentum exchange processes. All these processes are real phenomena and any
attempt for their description should be based on appropriate mathematical structures. The more than a century intensive interaction between differential geometry
and Lie group theory on one hand, and the theoretical and mathematical physics,
on the other hand, turned out to be exclusively useful, suggestive and creative
process. The author of the book under review has very successfully and in a very
appropriate way presented the idea that mathematics is not just abstract thinking
and physics is not just working in labs. It could be said that physics has always
stimulated mathematics to develop new branches, and mathematics has always
been very responsive and warm-hearted to the needs of physics. Starting with
Newton and Leibniz the greatest mathematicians of all times have contributed
considerably to deeply understanding and solving difficult physical problems.
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Book Review
The book under review consists of 22 chapters and two Appendices. Chapter 1
starts with a short review of some topological concepts introduces and illustrates
the concepts of smooth manifold and smooth maps of manifolds, and ends with
a description of smooth surfaces in R n . The following two chapters consider
the basic tangent structures (the tensor algebra on the manifold) and mappings of
tangent structures generated by mappings of manifolds. Among the illustrative
examples the metric tensor and related considerations are given more attention.
Chapter 4 is devoted to a detailed consideration of Lie derivative, its properties
and application in searching for isometries and conformal transformations. The
following four chapters five to eight consider the differential and integral calculus on manifold: exterior algebra and orientability (the Hodge star also), exterior
derivative, coderivative and corresponding Laplace – de-Rham operator, simplices
and chains and Stokes theorem, classical vector analysis. A short chapter on deRham cohomology (the Poincare lemma) follows. The next four chapters consider
the standard basic facts concerning Lie groups: left invariance, exponential map,
representations, intertwining operators, actions of Lie groups and Lie algebras
on manifolds, G-spaces. After a short chapter on Hamiltonian (symplectic) mechanics (including the moment map and reduction techniques) the considerations
continue mainly along the connection-curvature structures in vector and principal
bundles and their applications in physical theories – the so called gauge theories. Chapter 17 is devoted to the differential geometry of tangent and cotangent
bundles, Chapter 18 deals with Euler-Lagrange formulation of field theory, and
Chapter 22 considers Clifford algebra approach to spinors. Finally, something
that deserves to be specially noted: every chapter ends with a Summary text representing the most essential points and results of the chapter and all important
formulas. The first Appendix gives essentials on some algebraic structures and the
second Appendix (called “Starring”) gives a list of 83 names of mathematicians
and physicists with considerable contributions to the development of geometry
and physics. The Bibliography is limited to 31 names (and corresponding titles).
All basic material that is necessary for a young scientist in the field of geometrical formulation of physical theories is included and, which is more important, is
ordered and represented in a very appropriate manner, I would say, with a great
respect to the reader. Much of the material is given in the form of problems with
hints, so that the reader could test himself in the right understanding of what he
has already read. A physically educated reader will find, for example, a detailed
formal and actual explanation of the sense of Lie derivative (Chapter 4 and Section 8.2) and why it is so important as a powerful tool for symmetry considerations
of physical systems and how to find and study such symmetries. The role of local
Book Review
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isometries, i.e., Killing vector fields, on Riemannian manifolds, is also carefully
considered from the point of view how these vector fields are used in field theory
to generate integral conserved quantities like energy, momentum, angular momentum if the local stress-energy-momentum tensor is at hand.
For those who want to get into modern physical gauge theory the Chapters 1921 may serve as wonderful introduction. The basic geometrical concepts here
are the connection and the curvature, which are presented in a very clear way in
the both approaches: through splitting of the tangent bundles of some principal
G-bundles to horizontal and vertical subbundles (Chapters 19-20), and through
parallel transport and covariant derivative operator in vector bundles (the tangent
bundle is mainly considered in Chapter 15). In the principal bundle approach the
role of Frobenius integrability theorem is shown how it clarifies the most general sense of curvature as a measure of nonintegrability, which determines the
basic usage of curvature as adequate quantity describing interaction, i.e. energymomentum exchange, or force field, between physical systems (or subsystems
of a system). In my view, most of the physicists still do not pay due respect to
Frobenius integrability theorem as powerful adequate tool to be used in physical theories. For example, if a free physical system could be mathematically
considered as integrable distribution (in a definite sense), any interaction of its
two subsystems may be considered as nonintegrable (sub)distributions and the
corresponding curvatures to be used as measures of the corresponding internal
energy-momentum exchange. Although the soliton theory specialists feel glad
when succeed in representing some nonlinear evolution equation as “zero curvature condition”, the time-stability of the internal dynamical structure of the corresponding inter-related exchange processes (which are supposed to occur inside a
small spatial region) clearly suggests that in order to get a better understanding of
the corresponding spatially localized dynamical structure of the soliton, nonintegrability is so much important as integrability. This view clearly distinguishes the
primary importance of the curvature concept in describing the internal physical
energy-momentum exchange processes, since every physical process is necessarily accompanied by some energy-momentum space-time transport.
I truly believe that reading this book will bring a real pleasure to all physically
inclined young mathematicians and mathematically inclined young physicists,
moreover, I am fully convinced that getting appropriately acquainted with its contents really deserves. Therefore I would allow myself to characterize it as a very
good high level textbook and to recommend it to all young scientists being interested in finding correspondence between harmony in the physical world and
harmony in geometrical structures.
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Book Review
As a whole, (together with the exercises presented) the book is well written, very
well ordered and the exposition is very clear. It could serve as a good stand-by
introduction to modern differential geometry and to geometrical formulation of
physical theories. I would be glad to see in the second edition of the book a
chapter on Jet Bundles and their applications in mathematical physics.
Stoil G. Donev
Bulgarian Academy of Sciences
E-mail address: sdonev@inrne.bas.bg